Similarly for \(U=V^{-1}\)  the action on the derived category is the inverse of the spherical twist \(ST_\mathcal {O}\)  around the spherical object \(\mathcal {O}\) , which is given, following
Proposition 2.10 of [1]}, by the exact triangle
\(ST^{-1}_\mathcal {O}(E) \rightarrow E\xrightarrow{} \operatorname{Hom}(\operatorname{Hom}^\bullet _{K_{{P}^2}}(E,\mathcal {O}),\mathcal {O}) \xrightarrow{}\) 
